设 $z=\frac{1}{x} f\left(x^2 y\right)+x y g(x+y)$ ,其中 $f, g$ 具有二阶连续导数, 计算 $\frac{\partial^2 z}{\partial x^2}, \frac{\partial^2 z}{\partial x \partial y}$.
【答案】 由复合函数偏导数计算链式法则,有
$$
\begin{aligned}
\frac{\partial z}{\partial x}= & -\frac{2}{x^3} f\left(x^2 y\right)+\frac{2 y}{x} f_x\left(x^2 y\right)+y g(x+y)+x y g_x(x+y) \\
\frac{\partial^2 z}{\partial x^2}= & \frac{6}{x^4} f\left(x^2 y\right)-\frac{4 y}{x^2} f_x\left(x^2 y\right)-\frac{2 y}{x} f_x\left(x^2 y\right) \\
& +4 y^2 f_x\left(x^2 y\right)+2 y g_x(x+y)+x y g_{a x}(x+y ! \\
\frac{\partial^2 z}{\partial x \partial y}= & -\frac{2}{x} f_y\left(x^2 y\right)+\frac{2}{x} f_x\left(x^2 y\right)+2 x y f_{x y}\left(x^2 y\right)+g(x+y) \\
& +y g_y(x+y)+x g_x(x+y)+x y g_{s y}(x+y)
\end{aligned}
$$


系统推荐